TL;DR - Don’t worry about titles, instead aspire to be a better programmer.
Often those new to programming are confused about what exactly is the difference between a web developer, software developer, software engineer, etc. Here are some details on what set of skills these titles entail and what exactly the difference is.
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Web Developer - As a general rule web developers primarily work with websites, as opposed to building an application that would run on your computer (like Slack or Spotify, etc). If you are a web developer, you're probably working as either a frontend, backend or full stack developer (read What is a full stack developer to learn more).
Software Developer/Engineer - This title covers a larger range of skill sets / possible roles. While it can refer to designing, building, and maintaining software and applications, but it has also been used when referring to web developers. Often using the term “engineer” implies you have a deeper knowledge of computer science topics (such as data structures and algorithms).
The truth is that, there isn’t a lot of consistency with titles and you certainly shouldn’t be worrying about it too much. Good programmers are good programmers, no matter what special title they have. The specific set of expectations around a given title vary greatly depending on the company. More important you should learn what programming languages / technologies the company uses or you would be working with if you were to be hired for that role.
But if you really want to aspire to a specific title, you should strive to become a "benevolent dictator for life"!